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TEN PROBLEMS in HILBERT SPACE1 Dedicated to My Teacher

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TEN PROBLEMS in HILBERT SPACE1 Dedicated to My Teacher TEN PROBLEMS IN HILBERT SPACE1 BY P. R. HALMOS Dedicated to my teacher and friend Joseph Leo Doob with admiration and affection. TABLE OF CONTENTS PREFACE 887 1. CONVERGENT. Does the set of cyclic operators have a non-empty interior?... 889 2. WEIGHTED. Is every part of a weighted shift similar to a weighted shift?.... 892 3. INVARIANT. If an intransitive operator has an inverse, is its inverse also in­ transitive? 898 4. TRIANGULAR. Is every normal operator the sum of a diagonal operator and a compact one? 901 5. DILATED. Is every subnormal Toeplitz operator either analytic or normal?... 906 6. SIMILAR. Is every polynomially bounded operator similar to a contraction?.. 910 7. NILPOTENT. IS every quasinilpotent operator the norm limit of nilpotent ones? 915 8. REDUCIBLE. Is every operator the norm limit of reducible ones? 919 9. REFLEXIVE. Is every complete Boolean algebra reflexive? 922 10. TRANSITIVE. IS every non-trivial strongly closed transitive atomic lattice either medial or self-conjugate? 926 BIBLIOGRAPHY 931 PREFACE Machines can write theorems and proofs, and read them. The purpose of mathematical exposition for people is to communicate ideas, not theorems and proofs. Experience shows that almost always the best way to communicate a mathematical idea is to talk about concrete examples and unsolved problems. In what follows I try to communicate some of the basic ideas of Hilbert space theory by dis­ cussing a few of its problems. Nobody, except topologists, is interested in problems about Hil­ bert space; the people who work in Hilbert space are interested in problems about operators. The problems below are about operators. I do not know the solution of any of them. I guess, however, that A MS 1969 subject classifications. Primary 4702. Key words and phrases. Hilbert space, operators, cyclic vectors, weighted shifts, invariant subspaces, compact operators, dilations, contractions, quasinilpotence, reducibility, reflexive lattices, transitive lattices. 1 Lectures delivered at a Regional Conference in the Mathematical Sciences, sponsored by the National Science Foundation and arranged by the Conference Board of the Mathematical Sciences, at Texas Christian University, Fort Worth, Texas, May 25-29,1970. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 887 888 P. R. HALMOS [September they vary a lot in depth. Some have been around for many years and are known to be both difficult and important; others are untested and may turn out to be trivial. The problems are formulated as yes-or-no questions. That, I believe, is the only clear way to formulate any problem. What a mathematician usually wants to do is something vague, like study, determine, or characterize a class of objects, but until he can ask a clear-cut test question, the chances are he does not understand the problem, let alone the solution. The purpose of formulating the yes- or-no question, however, is not only to elicit the answer; its main purpose is to point to an interesting area of ignorance. A discussion of unsolved problems is more ephemeral than an expo­ sition of known facts. Deciding to make a virtue out of necessity, I chose to focus on a narrow segment of the present, rather than on a broad view of history. As often as possible the references are to recent publications, and the proofs presented in detail are more from cur­ rent folklore than from standard texts. Consequently some results (with appropriate credit lines to their discoverers) are published here for the first time. Most of the results and proofs with no assigned credit are "well known"; only a small number are mine. If a result is attributed to someone but is not accompanied by a reference to the bibliography, that means I learned it through personal communica­ tion, and, as far as I could determine, it is not available in print. The chief prerequisite for understanding the exposition is a knowledge of the standard theory of operators on Hubert spaces. In some cases that means that not even the spectral theorem is needed, and in other cases every available ounce of multiplicity theory would help to understand things clearly. In any case, the presentation is not cumulative; some readers who find §1 obscure may find §10 obvious. As an expository experiment I have put some exercises into each section. This is unusual in papers (as opposed to books). The reason I did it is that originally each section was a lecture, and lectures (as opposed to papers) can stand every bit of elasticity that anyone can think of to put in. The audience is trapped and cannot impatiently riffle the pages forward and back, and the lecturer can use something that is easily omitted or inserted without disturbing the continuity. As for terminology: Hubert space means a complete, complex, inner-product space; subspace means a closed linear manifold; operator means a bounded linear transformation. The Hubert spaces of principal interest are the ones that are neither too large nor too small, i.e., they are separable but infinite-dimensional. The problems are stated without any such explicit size restrictions, but in some License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1970] TEN PROBLEMS IN HILBERT SPACE 889 cases they become trivial for either the large spaces, or the small ones, or both. I thank Peter Fillmore, Pratibha Gajendragadkar, and Eric Nordgren for reading the first written version of these lectures and for making many helpful suggestions. 1. CONVERGENT Problem L Does the set of cyclic operators have a non-empty interior? A vector ƒ in a Hilbert space H is cyclic for an operator A on H in case the smallest subspace of H that contains ƒ and is invariant under A is H itself. An operator is cyclic in case it has a cyclic vector. In the finite-dimensional case, A is cyclic if and only if its minimal polynomial is equal to its characteristic polynomial. (Equivalently: each eigenvalue occurs in only one Jordan block, or the space of eigenvectors corresponding to each eigenvalue has dimension 1 [24, p. 69 et seq]; cf. also [33].) It follows easily that in the finite-dimen­ sional case the set 6 of cyclic operators is open. If the dimension is n, then the operators with n distinct eigenvalues constitute a dense set; it follows that 6 is dense. (Consequence: C is not closed; for instance, diag(l, l + lA*>-*diag(l, 1).) There is only one reasonable topology for the operators on a finite-dimensional space; in the infinite-dimensional case there are several. Problem 1 refers to the norm (or uniform) topology, the one according to which An-*A in case \\An—A\\—»0. If ƒ is a cyclic vector for A, then the countable set {ƒ, Af> A2f, • • •} spans H ; it follows that in the non-separable case C is empty. In the separable infinite-dimensional case what little is known is negative. The set e is surely not open; the cyclic operator diag(l, i, i, • • • ) is the limit of the non-cyclic operators diag(l, J, • • • , l/n> 0, 0, 0, • • • ). It is, of course, possible that the interior of G is non-empty; that is what Problem 1 is about. A reason­ able candidate for an operator in the interior of 6 is the unilateral shift U. (According to the most easily accessible definition U is the 2 operator on I that sends (£0, £i, £2, • • • ) onto (0, £0, £1, £2, • • • ).) That candidate happens to fail (J. G. Stampfli). The set 6 is also not dense; in face if V is the direct sum of two copies of U and || F—^4|| <1, then A is not cyclic. Indeed: ||l - A*V\\ = ||7*7 - A*V\\ â ||F* - A*\\ < 1, so that A* V is invertible; it follows that V*A is inverti ble and hence that ran A C\ ker F*=0. Since, thus, ran A is disjoint from the 2- dimensional space ker V*t the co-dimension of ran A must be at least License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 890 P. R. HALMOS [September 2. (The proof so far was needed to get this inequality only. An alter­ native way to get it is to note that it is a special case of a small part of the theory of Fredholm operators [35].) Now it is clear that A cannot be cyclic ; for each ƒ, the span of {ƒ, Af, A 2f, • • •} has co-dimension at least 1. The point of the preceding discussion is to call attention to the existence of topological problems of analytic importance in operator theory. There are many; Problem 1 is just a sample. Other examples are Problems 7 and 8, about the closures of certain sets of operators, or, in other words, about approximation theory. Here is a well known example of the same kind: is the set of all invertible operators dense? The answer is yes for finite-dimensional spaces, and furnishes an occasionally useful proof technique; to prove something for all matrices, prove it for the non-singular ones first and then approxi­ mate. For infinite-dimensional spaces the answer is no [20, Probl em 109]. It is slightly less well known that the answer remains no even for a somewhat larger set, the set of kernel-free operators. (If A is invertible, then A is kernel-free, i.e., ker 4=0, but the converse is not true.) In other words, there exists a non-empty open set every element of which has a non-trivial kernel.
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